A COMPUTATIONAL MATHEMATICAL FRAMEWORK FOR HIGH-DIMENSIONAL ENGINEERING DATA ANALYSIS USING ADVANCED LINEAR ALGEBRA, MATRIX FACTORIZATION AND OPTIMIZATION TECHNIQUES

Abstract

High-dimensional engineering datasets are increasingly generated from smart sensors, simulation platforms, industrial monitoring systems, communication networks, and intelligent control environments. However, the large number of variables, nonlinear relationships, redundant features, and computational complexity often reduce the efficiency and accuracy of conventional data analysis methods. This study presents a mathematical framework for high-dimensional engineering data analysis using advanced linear algebra and optimization techniques. The proposed framework integrates matrix decomposition, vector space transformation, dimensionality reduction, eigenvalue-based feature representation, convex optimization, gradient-based learning, and regularization methods to improve data processing, predictive modeling, and machine learning performance in engineering applications. The methodology focuses on transforming complex engineering datasets into optimized mathematical representations by applying Principal Component Analysis, Singular Value Decomposition, least-squares optimization, and regularized regression techniques. These methods help reduce noise, remove irrelevant features, enhance computational speed, and improve model interpretability. The optimized feature space is then used with machine learning models such as Support Vector Machine, Random Forest, and Neural Network classifiers for predictive analysis and decision support. Experimental results demonstrate that the proposed mathematical framework significantly improves model performance compared with traditional feature-processing methods. The dimensionality of the dataset was reduced by 42.6%, while preserving 96.8% of the original data variance. The proposed framework achieved an overall prediction accuracy of 97.3%, precision of 96.5%, recall of 95.9%, and F1-score of 96.2%. In addition, computational training time was reduced by 31.4%, and mean squared error decreased from 0.084 to 0.031 after applying optimization-based feature transformation. The results confirm that advanced linear algebra and mathematical optimization techniques provide a strong foundation for high-dimensional engineering data analysis. Overall, this research highlights the importance of mathematical modeling in improving machine learning efficiency, predictive accuracy, and intelligent decision support for modern engineering systems. The proposed framework can be applied in areas such as smart manufacturing, structural health monitoring, electrical systems, robotics, and engineering design optimization